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Multi-peak positive solutions to a class of Kirchhoff equations
- Publication Year :
- 2017
- Publisher :
- arXiv, 2017.
-
Abstract
- In the present paper, we consider the nonlocal Kirchhoff problem$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$ where a, b>0, 1<p0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.
- Subjects :
- Class (set theory)
Singular perturbation
Partial differential equation
General Mathematics
Mathematical analysis
Function (mathematics)
01 natural sciences
Kirchhoff equations
010305 fluids & plasmas
Term (time)
Reduction (complexity)
Mathematics - Analysis of PDEs
0103 physical sciences
35A01 35B25 35J20 35J60
FOS: Mathematics
010306 general physics
Lyapunov–Schmidt reduction
Mathematics
Analysis of PDEs (math.AP)
Subjects
Details
- ISSN :
- 03082105
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....18e830dbc099847f245e157291e0264c
- Full Text :
- https://doi.org/10.48550/arxiv.1708.01770